Gases

This is Unit Seven of Chemistry, Part Two: effusion, kinetic-molecular theory and real gases. Part One built the gas laws empirically — four relationships found by measurement, combined into one equation, with two assumptions noted and set aside. This part does two things with that. First it explains the laws, by building a model of what gas molecules are actually doing, and the book is deliberate that this model comes after the laws rather than before, because it exists to account for them. Then it returns to the two assumptions and asks where they fail. By the end you will be able to say when reacting gas volumes stand in for mole ratios; tell diffusion from effusion; say which gas effuses faster and why the intuition runs backwards; say what kinetic-molecular theory is for; and say when a real gas deviates most from ideal behaviour.

Two entries. When a balanced equation's coefficients can be applied straight to measured volumes, and the fuller procedure for when the conditions do not allow that shortcut.

Volumes as ratios.

Chem 1283 Gas Volumes in Stoichiometric Ratios

At the same temperature and pressure, reacting gases' volumes follow the same whole-number ratio as their equation coefficients — a consequence of Avogadro's law, since equal volumes then hold equal moles. The shortcut holds only when every gas is at identical temperature and pressure; mixed conditions require converting to moles first.

When they differ.

Chem 1286 Gas Stoichiometry from Reaction Volumes

Solving a gas stoichiometry problem means converting a given volume to moles with the ideal gas law when conditions are unmatched, applying the balanced equation's mole ratio, then converting back to volume under the target conditions — or using the volume ratio directly when conditions already match. Applying the equation's mole-ratio numbers straight to volumes measured at two different temperatures or pressures skips the conversion the shortcut depends on.

Three entries describing what gas molecules do. Two words for spreading that are often used interchangeably and should not be, and then the average distance a molecule covers before something gets in its way.

Spreading out.

Chem 1291 Diffusion

Diffusion is the spread of gas molecules from a region of higher concentration to one of lower concentration through random motion. No molecule moves toward low concentration on purpose — the net drift emerges from millions of individually random directions, and there are more molecules available to wander out of the crowded region than into it.

Escaping through a hole.

Chem 1294 Effusion

Effusion is the escape of gas molecules through a tiny opening into a vacuum, one molecule passing through at a time. It is diffusion's narrower cousin: diffusion spreads through open space, while effusion specifically crosses a small opening into a vacuum.

Between collisions.

Chem 1297 Mean Free Path

The mean free path is the average distance a gas molecule travels between successive collisions with other molecules. It is an average over enormous numbers of collisions — no single molecule travels exactly that distance every time, and the spread around it is wide.

Three entries. The relationship between a gas's mass and how fast it escapes, an intuition that gets it exactly backwards, and the calculation — which has a second trap in it about time against rate.

Inversely, by square root.

Chem 1300 Graham's Law of Effusion

A gas's rate of effusion is inversely proportional to the square root of its molar mass. The ratio of two gases' effusion rates therefore scales with the square root of the inverse molar-mass ratio, not with molar mass directly — which is a milder dependence than a direct ratio would give.

Backwards intuition.

Chem 1302 The Direct-Proportionality Misreading of Graham's Law

Assuming effusion rate scales directly with molar mass, rather than inversely with its square root, misreads Graham's law and reverses which gas escapes faster. The inverse-square-root relationship means the lighter gas always effuses faster, never the heavier one — the opposite of what a picture of heavier molecules carrying more momentum suggests.

Rate, or time.

Chem 1305 Calculating Relative Effusion Rates

Calculating the ratio of two gases' effusion rates means taking the square root of the inverse ratio of their molar masses — or, for effusion time, inverting that same ratio, since time and rate vary oppositely. Using molar mass directly instead of its square root, or forgetting that a longer effusion time means a slower rate, produces a ratio off by more than a rounding error.

Six entries, and the structural hinge of the unit. Everything so far has been empirical — relationships found by measurement. This is the model that explains why they hold, built deliberately after them rather than before, and once it exists it derives Graham's law too.

Four postulates.

Chem 1308 Kinetic-Molecular Theory's Core Postulates

Kinetic-molecular theory models a gas as tiny particles in constant, random motion that collide elastically and exert no force between collisions. Those postulates map directly onto the two assumptions the ideal gas law already made — zero particle volume and zero intermolecular force. The theory explains why the gas laws hold; it is not a separate topic sitting alongside them.

Why each law holds.

Chem 1311 Explaining the Gas Laws from Kinetic-Molecular Theory

Explaining a gas law from kinetic-molecular theory means tracing how a change in temperature, volume or amount alters the frequency and force of particle-wall collisions, then showing that traced change reproduces the law's known relationship. Halving a container's volume doubles collision frequency, which is Boyle's law. Treating the theory as a fifth independent gas law, rather than the mechanism behind the first four, skips the entire point of building it afterwards.

Energy, not speed.

Chem 1315 Average Kinetic Energy and Temperature

A gas sample's average molecular kinetic energy is directly proportional to its kelvin temperature, regardless of the gas's identity — so two different gases at the same temperature have the same average kinetic energy. Equal average kinetic energy does not mean equal average speed: since energy depends on mass as well as speed, the lighter gas must be moving faster.

A spread of speeds.

Chem 1318 Maxwell-Boltzmann Speed Distribution

At a fixed temperature, a gas sample's molecular speeds spread across a range, peaking at a most-probable speed — the Maxwell-Boltzmann distribution. Not every molecule moves at the average: the distribution's whole shape, rather than any single number, describes the gas, and raising temperature both shifts the peak and widens the spread.

One representative speed.

Chem 1321 Calculating Root-Mean-Square Speed

Calculating a gas's root-mean-square speed means substituting its molar mass in kilograms and the kelvin temperature into the square root of three R T divided by molar mass. Leaving molar mass in grams per mole, instead of converting to kilograms per mole to match R's joule-based units, produces a speed far larger than the correct value.

Graham's law, derived.

Chem 1325 Kinetic-Molecular Theory Explains Graham's Law

Since effusion rate tracks molecular speed, and lighter molecules move faster at equal kinetic energy, kinetic-molecular theory derives Graham's inverse-square-root relationship directly. That derivation shows Graham's law follows from the same postulates as the other gas laws — not a separate coincidence that happens to hold.

Six entries closing the unit. Part One noted two assumptions behind the ideal gas law — that particles have zero volume and exert no forces on each other — and set them aside. This is where both fail: where the failure shows, how it is measured, how it is corrected, and then the capstone running one sample through the whole chain.

Both assumptions fail.

Chem 1330 Real Gases Deviate from Ideal Behavior

Real gas molecules do have volume and do attract each other — the two things the ideal gas law assumed away — so real gases deviate most from ideal predictions at high pressure and low temperature, where molecules are crowded and slow. Compressed hard, carbon dioxide occupies less volume than the ideal law predicts, because its molecules pull on one another. Real gases approximate ideal behaviour at low pressure and high temperature: deviation is a matter of degree, not all-or-nothing.

Measuring the gap.

Chem 1333 The Compressibility Factor

The compressibility factor is the ratio of a gas's actual molar volume to the volume the ideal gas law predicts at the same conditions. A factor of exactly one signals ideal behaviour. Below one means attraction dominates, shrinking the gas below its ideal volume; above one means molecular volume dominates instead — so the factor says which assumption is failing, not only that one is.

When ideal works.

Chem 1336 Conditions Favoring Ideal Behavior

A real gas behaves most nearly ideally at low pressure and high temperature, where molecules sit far apart and move too fast for attraction to matter much. Both conditions are required: a cold, low-pressure gas can still deviate if it is near condensing, since low temperature lets attraction act however far apart the molecules are.

Two corrections.

Chem 1339 The van der Waals Equation

The van der Waals equation corrects the ideal gas law with two added terms — one for molecular size, one for intermolecular attraction — matching the two assumptions that failed. Both correction terms shrink toward zero as volume grows, so at large volumes the equation reduces to the ideal gas law itself.

How big is the correction.

Chem 1343 Calculating Pressure with the van der Waals Equation

Calculating a real gas's pressure means substituting moles, volume, temperature and that gas's own size and attraction constants into the corrected equation, then comparing against the ideal gas law's prediction for the same sample. Expecting a large gap at every condition ignores that the correction is small whenever pressure is not very high and temperature is not very low — which is most ordinary conditions.

One flask, four stages.

Chem 1350 Capstone: From Manometer Reading to Real-Gas Correction

A sealed flask of carbon dioxide reads 748 millimetres of mercury on a closed-end manometer — directly the gas pressure, since that instrument references a vacuum — which converts to 0.984 atmospheres. With 1.56 grams in one litre at 323 kelvin, the ideal gas law gives about 0.0372 moles, so a molar mass near 41.9 against carbon dioxide's known 44.0, close enough to identify the gas from measurement alone. Applying van der Waals with carbon dioxide's own constants shifts the pressure only slightly, because one litre at 323 kelvin sits far from where the correction matters. Treating these as four unrelated procedures is the fragmented reading the capstone exists to correct.

Five things to carry out of this part, and out of the unit. Reacting gas volumes stand in for mole ratios only when every gas is at the same temperature and pressure — that is Avogadro's law doing the work, and mixed conditions require converting to moles first. Diffusion is spreading through open space from high concentration toward low; effusion is the narrower case of escaping through a small opening into a vacuum, and neither involves any molecule aiming anywhere. The lighter gas always effuses faster, because rate goes inversely with the square root of molar mass — the intuition that heavier molecules should push through faster reverses the relationship. Kinetic-molecular theory exists to explain the gas laws rather than to sit beside them as a fifth: its postulates about particles in constant random motion reproduce all four empirical laws by tracing collision frequency and force, and they derive Graham's law too. And a real gas deviates most from ideal behaviour at high pressure and low temperature, where its molecules are crowded enough for their own volume to matter and slow enough for their mutual attraction to act — which is exactly where the ideal gas law's two assumptions fail. At low pressure and high temperature both assumptions hold well enough that the ideal law is the right tool. Unit Eight takes up liquids, solids and solutions.